Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \(\begin{array}{ll}\text { 25. }(7,0) & m=1\end{array}\) \((0,-4) \quad m=-1\)

Step-by-Step Solution

Answer

The equation of the first line is \(y = x - 7\) and the equation of the second line is \(y = -x - 4\)

Step 1: Solving for the first line

For the first line with point (7,0) and slope \(m = 1\), use the point-slope formula \(y - y1 = m(x - x1)\), and put in the point's coordinates for \(x1\) and \(y1\). We get \(y - 0 = 1*(x - 7)\). Simplifying this yields the equation \(y = x - 7\). The sketch of this line starts at the point (7,0) and rises with a slope of 1.

Step 2: Solving for the second line

For the second line with point (0,-4) and slope \(m = -1\), we use the point-slope formula again. This gives us \(y - (-4) = -1*(x - 0)\), simplifying this gives the equation \(y = -x - 4\). The sketch of this line starts at the point (0,-4) and goes down with a slope of -1.

Key Concepts

Point-Slope Formula

The point-slope formula is a valuable tool in algebra for constructing the equation of a line that passes through a given point and has a specified slope. The formula is represented as:

• \(y - y_1 = m(x - x_1)\)

Here, \((x_1, y_1)\) are the coordinates of the known point and \(m\) is the slope of the line. This formula helps transition from known facts about a line — a point and a slope — to its algebraic equation.

The point-slope form is particularly handy when you know a point on the line and the slope but are not sure about the intercepts. It makes using information straightforward to get the line's equation.

• Example 1: For a line passing through (7, 0) with a slope of 1, the formula gives \(y - 0 = 1(x - 7)\), simplifying to \(y = x - 7\).
• Example 2: For a line passing through (0, -4) with a slope of -1, the formula is \(y + 4 = -1(x - 0)\), simplifying to \(y = -x - 4\).

Graphing Lines

Graphing lines involves plotting points on a coordinate plane and drawing a straight path that represents the equation of the line. Understanding how to graph lines effectively requires a basic grasp of how the slope and intercepts influence the line's path.

To graph a line given by its equation, follow these steps:

• Identify a point on the line. The easiest starting point is often the y-intercept, found when \(x = 0\).
• Use the slope to find another point. The slope \(m\) tells you the direction and steepness of the line: if \(m = 1\), the line rises one unit up for every one unit it moves right; if \(m = -1\), it moves one unit down for each unit right.
• Connect these points with a straight line.

Graphing is a visual way to understand how lines behave and interact on the plane, and it's a reliable method to verify your algebraic solutions.

In practice, sketching the line that passes through (7, 0) with a slope of 1 helps visualize a line tilting upwards, starting from (7, 0) and continually ascending. Similarly, the line through (0, -4) with a slope of -1 represents a downward tilt, starting at (0, -4) and descending to the right.

Slope-Intercept Form

The slope-intercept form of a line is a friendly and versatile way to express linear equations. It's commonly written as:

• \(y = mx + b\)

In this formula, \(m\) is the slope, and \(b\) is the y-intercept, or the point where the line crosses the y-axis.

This format makes it simple to graph lines because it directly reveals key features of the line:

• The slope \(m\) indicates how steep the line is.
• The y-intercept \(b\) gives a specific point on the graph, making it an automatic starter point for graphing.

The slope-intercept form provides a quick way to identify the behavior of the line and is especially useful in comparing different lines.

For example, from our exercise, the equation \(y = x - 7\) is already in slope-intercept form with a slope of 1 and a y-intercept of -7. The line \(y = -x - 4\) shows a slope of -1 and a y-intercept of -4. These simplified forms make it easy to sketch these lines accurately on a graph.